Stationary Points & Turning Points (AQA GCSE Further Maths): Revision Note

Stationary points & turning points

What are stationary points?

• A stationary point is any point on a curve where the gradient is zero

• To find stationary points of a curve

Step 1
Find the first derivative $\frac{dy}{dx}$

Step 2
Solve $\frac{dy}{dx}=0$ to find the $x$-coordinates of any stationary points

Step 3
Substitute those $x$-coordinates into the equation of the curve to find the corresponding $y$-coordinates

• A stationary point may be either a local minimum, a local maximum, or a point of inflection 

Stationary points on quadratics

• The graph of a quadratic function only has a single stationary point

• For a positive quadratic this is the minimum; for a negative quadratic it is the maximum

  • No need to talk about 'local' here, as it is the overall minimum/maximum for the whole curve

• The y value/coordinate of the stationary point is therefore the minimum or maximum value of the quadratic function

• For quadratics especially, minimum and maximum points are often referred to as turning points

How do I determine the nature of stationary points on other curves?

• For a graph there are two ways to determine the nature of its stationary points 

• Method A

  • Compare the signs of the first derivative, $\frac{\mathrm{d}y}{\mathrm{d}x}$, (positive or negative) a little bit to either side of the stationary point

  • (After completing Steps 1 - 3 above to find the stationary points)

Step 4
For each stationary point find the values of the first derivative a little bit 'to the left' (i.e. for a slightly smaller x value) and a little bit 'to the right' (i.e. for a slightly larger x value) of the stationary point

Step 5
Compare the signs (positive or negative) of the derivatives on the left and right of the stationary point

• If the derivatives are negative on the left and positive on the right, the point is a local minimum (a u-shape)

• If the derivatives are positive on the left and negative on the right, the point is a local maximum (an n-shape)

• If the signs of the derivatives are the same on both sides (both positive or both negative) then the point is a point of inflection (a shape)   

• Method B

  • Look at the sign of the second derivative (positive or negative) at the stationary point

  • (After completing Steps 1 - 3 above to find the stationary points)

Step 4
Find the second derivative $\frac{{\mathbf{d}}^{\mathbf{2}}\mathit{y}}{\mathbf{d}{\mathit{x}}^{\mathbf{2}}}$

Step 5
For each stationary point find the value of $\frac{{\mathbf{d}}^{\mathbf{2}}\mathit{y}}{\mathbf{d}{\mathit{x}}^{\mathbf{2}}}$ at the stationary point
i.e. substitute the x-coordinate of the stationary point into $\frac{{\mathbf{d}}^{\mathbf{2}}\mathit{y}}{\mathbf{d}{\mathit{x}}^{\mathbf{2}}}$ and evaluate

• If $\frac{{\mathbf{d}}^{\mathbf{2}}\mathit{y}}{\mathbf{d}{\mathit{x}}^{\mathbf{2}}}$ is positive then the point is a local minimum

• If $\frac{{\mathbf{d}}^{\mathbf{2}}\mathit{y}}{\mathbf{d}{\mathit{x}}^{\mathbf{2}}}$ is negative then the point is a local maximum

• If $\frac{{\mathbf{d}}^{\mathbf{2}}\mathit{y}}{\mathbf{d}{\mathit{x}}^{\mathbf{2}}}$ is zero then the point could be a local minimum, a local maximum OR a point of inflection

  • use Method A to determine which